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We construct a series of homomorphisms from the $Y$-filtration on the monoidof homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction ofthe LMO functor. The restriction of our homomorphism to the lower centralseries of the Torelli group does not factor through Morita's refinement of theJohnson homomorphism. We use it to show that the abelianization of the Johnsonkernel of a closed surface has torsion elements. We also determine the thirdgraded quotient $Y_3\mathcal{IC}_{g,1}/Y_4$ of the $Y$-filtration.

Abelian quotients of the $Y$-filtration on the homology cylinders via the LMO functor